Whisk together the milk, yeast, and a teaspoon of the sugar. Set it aside until it blooms and becomes foamy. This will take 5-10 minutes.
While waiting for the yeast to bloom, whisk together the flour, mahlepi, mastic, salt, orange zest, and sugar in the bowl of a stand mixer. You can also make this bread by hand if you don't have a mixer.
Now, add in the bloomed yeast mixture, along with the vanilla extract and honey. Mix on low speed until the mixture comes together and pulls away from the sides of the bowl. If you do not have a mixer, bring all of the ingredients together in a bowl with your hands. Once a cohesive ball has formed, turn it onto your work surface. Whether by hand or with a mixer, begin working in the butter a ½ tablespoon at a time, adding in the next addition of butter once the previous addition has been fully incorporated. If using a KitchenAid mixer, use speed 2.
Once the butter has been fully incorporated, continue to knead the dough for 10-15 minutes on speed 2 or by hand (by hand may take 5-10 minutes longer). Do this until you can pass the windowpane test, which is where you stretch a piece of dough and if it doesn't break and you can see light through it, then the gluten has properly developed.
Very lightly oil a large bowl. Form the dough into a ball and place it into the oiled bowl, turning the dough over to make sure all sides are covered in oil. Cover the bowl with plastic wrap and put it in a warm spot to rise until it has doubled in size. This may take 1-2 hours depending on the temperature of your kitchen. Enriched doughs like this tend to rise slowly.
Punch the risen dough down and tip it onto your work surface. Divide it in half. Then, divide each half into three pieces. Take each piece and roll it into long snake and then fold in half and roll it out again. Repeat this two to three times. This step is necessary to create the classic tsoureki texture, which has long elastic threads in the dough. Once you've done this to all 6 pieces, place them under plastic wrap and let them rest for 10 minutes to allow the gluten to relax for final shaping.
For the final shaping, roll out three pieces of dough until they are each about 12-15 inches long. Then, braid the dough into a classic braid shape. Tuck the ends under the braid to make a tidy looking loaf. Repeat this same process for the remaining three pieces of dough. Cover them and let them rise for 30 minutes. Do not let them double in size. You just want them to be lightly puffed.
Brush them with an egg wash and sprinkle them with sliced almonds. Alternatively, you could sprinkle them with sesame seeds.
Bake them in a 375℉ / 190℃ pre-heated oven for 25-30 minutes, or until they're a deep golden brown. Let them cool on a rack completely before serving. If you have time, let them sit in a covered container for a day before serving. This will intensify the flavor of the spices.